Journal of Structural Geology

7/29/2019 Journal of Structural Geology

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Journal of Structural Geology,Vol. 15, Nos 3-5, pp. 413 to 422, 1993 019l--8141/93 $06.00+ 0.00Printed in Grea t Britain ~ 1993 Pergamon Press Ltd

T h e d e f o r m a t i o n m a t r ix f o r s i m u l t a n e o u s s i m p l e s h e a r i n g , p u r e s h e a r i n ga n d v o l u m e c h a n g e , a n d i t s a p p l ic a t io n t o t r a n s p r e s s i o n - t r a n s t e n s i o ntec tonicsH A A K O N F O S S E N * a n d B A S I L T I K O FF

D e p a r t m e n t o f G e o l o g y a n d G e o p h y s i c s , U n i v e r s it y o f M i n n e s o t a , M i n n e a p o l i s , M N 5 5 4 5 5 , U . S . A .(Received 23 December1991; accepted in revised orm 22 July 1992)

A b s t r a c t - - S i m u l t a n e o u s s i m p l e s h e a r in g a n d p u r e s h e a r i n g , w it h o r w i t h o u t a d d i t i o n a l v o l u m e c h a n g e , c a n b ec o m b i n e d i n t o a s i n g l e , u p p e r t r i a n g u l a r d e f o r m a t i o n m a t r ix . T h e o f f -d i a g o n al t e r m , F , i s n a m e d t h e e f f e c ti v es h e a r s t r a i n , a n d i s a s i m p l e f u n c t io n o f t h e p u r e s h e a r i n g a n d s i m p l e s h e a r i n g c o m p o n e n t s . A t h r e e - d i m e n s i o n a ld e f o r m a t i o n m a t r i x f o r t h e s i m u l t a n e o u s c o m b i n a t i o n o f c o a x i al d e f o r m a t i o n , w i t h o r w i t h o u t a d d i t io n a l v o l u m ec h a n g e , a n d u p t o t h r e e s i m p l e s h e a r i n g s y s te m s w i t h m u t u a l ly o r t h o g o n a l s h e a r p l a n e s i s a ls o p r e s e n t e d . B yus i ng t h i s m a t r i x , one can eas i l y ex t r ac t t he va r i ous p r ope r t i e s o f i nc r em en t a l a s we l l a s f i n i t e s t r a i n , and t hep r og r ess i ve a s we l l a s fi n i t e r o t a t i on o f pass i ve m ar ke r s du r i ng de f o r m a t i on .T h e c a s e o f t r a n s p r e s s i o n - t r a n s t e n s i o n is r e v i se d , u s i n g t h e u n i f ie d d e f o r m a t i o n m a t r ix . T h e o r i e n t a t i o n o f t h em a j o r ax i s o f t he s t r a i n e l l i p so i d 0q ) i s a l ways ho r i zon t a l i f t he de f o r m a t i on i s t r an s t ens i ona l , sw i t ches f r omhor i zon t a l to ve r t i ca l du r i ng t r ansp r ess i on a l wr ench i ng (1 > Wk > 0 .81 f o r cons t an t vo r t i c i ty de f o r m a t i on s ) , andi s a lways ve r t i ca l f o r h i gh l y t r ansp r ess i o na l de f o r m a t i ons ( Wk ~< 0 .81 ) . F o r t r ans p r ess i on , m a t e r i a l l i nes i n i ti a l lyr o t a t e t owar ds t he ho r i zon t a l shea r i ng d i r ec t i on , bu t gene r a l l y t u r n t o r o t a t e t owar ds t he ve r t i ca l ax i s a f t e r ace r t a i n s t r a i n . F o r t r ans t ens i on , a l l m a t e r i a l l i nes r o t a t e t owar ds a d i r ec t i on i n t he ho r i zon t a l p l ane wh i ch i sob l i que t o t he shea r i ng d i r ec t i on .

I N T R O D U C T I O NIT IS s i m p l e a n d u s e fu l t o t h i n k o f ro c k d e fo rm a t i o n i nt e r m s o f s i m p l e s h e a r , p u r e s h e a r a n d v o l u m e c h a n g e(d i l a t io n ) , a n d t h e s e d e fo rm a t i o n s h a v e f r e q u e n t l y b e e na p p l i e d t o e x p l a i n d e fo rm a t i o n s t ru c t u re s o b s e rv e d i nro c k s . E v e n u n d e r f a i r l y s i m p l e c o n d i t i o n s , h o we v e r ,p l a n e s t r a i n d e fo rm a t i o n i s l i k e ly t o b e a c o m b i n a t i o n o ft w o o r m o r e o f th e s e c o m p o n e n t s . F o r i n s t a nc e , c o m b i -n a t i o n s o f a s i m p l e s h e a r a n d v o l u m e c h a n g e a r e p ro b -a b l y v e r y c o m m o n , e v e n i n ' p e r f e c t' s h e a r z o n e s o fc o n s t a n t th i c k n e ss a n d w i t h u n d e f o r m e d o r h o m o g e n e -o u s l y d e f o r m e d w a l l r o c k s ( R a m s a y & G r a h a m 1 97 0) .M o r e g e n e r a l s h e a r z o n e s d o n o t o b e y t h e s e c o n st r a in t s,a n d h a v e a d d i t io n a l c o m p o n e n t s o f p u r e s h e a r . H e n c e ,s i m p l e s h e a r , v o l u m e c h a n g e a n d p u re s h e a r a r e o n l ye n d - m e m b e r s o f a w i d e ra n g e o f d e f o r m a t i o n t y p e s , a n de a c h a l o n e c a n r a r e l y e x p l a in t h e d e fo rm a t i o n s t ru c t u re so b s e rv e d i n ro c k s . T h e a c k n o wl e d g m e n t o f t h i s f a c t l e dto a se r ies o f funda me n ta l a r t i c les in the 1980s wh icht r e a t e d f i n it e d e fo rm a t i o n a s a c o m b i n a t i o n o f s i m p l es h e a r , p u r e s h e a r a n d / o r v o l u m e c h a n g e ( R a m s a y 1 9 80 ,C o wa rd & Ki m 1 9 8 1 , K l i g f i e l d e t a l . 1981 , Sanderson1 9 8 2 , C o wa rd & P o t t s 1 9 8 3 , S a n d e r s o n & M a rc h i n i1 98 4 ). A m a t h e m a t i c a l l y c o n v e n i e n t o rd e r o f s u p e rp o s i -t i o n o f s i m p l e s h e a r , p u re s h e a r a n d / o r v o l u m e c h a n g ewa s a s s u m e d i n t h e s e w o rk s t o m o d e l f i n it e s tr a i n . T h eo r d e r o f s u p e r p o si t io n w a s c h o s e n w i t h o u t r e f e r e n c e t ot h e a c t u a l g e o l o g i c a l d e fo r m a t i o n h i s t o ry , wh i c h wa s n o tc o n s i d e re d i n m o s t o f t h e s e a r t ic l e s.I t is r e a s o n a b l e t o a s s u m e , a n d i n s o m e c a s e s i t c a n b ed e m o n s t r a t e d ( e . g . P a s s c h i e r & Ura i 1 9 8 8, W a l l is 1 9 92 ) ,

* P r e s e n t a d d r e s s : S t a t o i l, D D B , 5 0 20 B e r g e n , N o r w a y .

t h a t n a t u ra l , p ro g re s s i v e d e fo rm a t i o n g e n e ra l l y o c c u r sb y s i m u l t a n e o u s p u re s h e a r i n g , s i m p l e s h e a r i n g , a n d / o rd i la t in g (p ro g re s s i v e v o l u m e c h a n g e ) . (T h e - i n g suffixwi l l b e u s e d wh e re i t i s e s s e n t i a l t o e m p h a s i z e t h ek i n e m a t ic a s p e c t o f d e f o r m a t i o n , a s s u g g e s t ed b y M e a n s1 9 90 , a n d n o t m e re l y t h e f i n it e s t a te o f d e fo rm a t i o n . )He n c e , i f d e fo rm a t i o n h i s t o ry is a c o n c e rn , a n a p p ro a c hd i f f e r e n t f ro m t h a t o f d i sc r e t e s t r a in f a c t o r i z a t i o n i sr e q u i r e d . T h e t h e o ry n e e d e d i s a v a i l a b l e i n a c o n t i n u u mm e c h a n i c s f r a m e wo rk ( e . g . M a l v e rn 1 9 6 9 ) , a n d i s e l e -g a n t l y la i d o u t b y R a m b e r g (1 9 7 5 ) in t e rm s o f p u res h e a r i n g a n d s i m p l e s h e a r i n g s t r a i n r a t e s . Ho we v e r ,s i n c e m a n y g e o l o g i s t s a r e n o t v e ry f a m i l i a r w i t h t h et h e o r y o f c o n t in u u m m e c h a n i c s, w e p r e s e n t a c o n -v e n i e n t w a y o f c o m b i n i n g t h e s e t h r e e d e f o r m a t i o n e n d -m e m b e r s i n t o a s i n g l e , u n i f i e d d e fo rm a t i o n m a t r i x i nt e rm s o f s i m p l e sh e a r i n g a n d p u re s h e a r i n g -d i l a t i n gc o m p o n e n t s . W e a l s o e x t e n d t h e d i s c u s s i o n t o t h r e ed i m e n s i o n s i n wh i c h p u re s h e a r i n g b e c o m e s t h r e e -d i m e n s i o n a l c o a x i a l d e fo rm a t i o n , a n d i n wh i c h s e v e ra lo r t h o g o n a l s e t s o f s i m u l t a n e o u s s i m p l e s h e a r in g d e fo r -m a t i o n s m a y o c c u r . T h e a p p l i c a t io n o f th i s t h e o ry t os t ru c t u ra l m o d e l l i n g i s f i n a l l y d e m o n s t r a t e d fo rt r a n s p re s s i o n a l - t r a n s t e n s i o n a l d e fo rm a t i o n .

THE DEFORMATION MATRIXF o r h o m o g e n e o u s d e f o r m a t i o n , t h e m a t r i x D d e -

s c r i b e s a li n e a r t r a n s fo rm a t i o n r e l a t in g t h e u n d e fo r m e dv e c t o r o r p o i n t ( x ) i n a C a r t e s i a n c o -o rd i n a t e s y s t e m t oi ts p o s it i o n a f t e r d e fo rm a t i o n (x ' ) :x ' = Dx (1 )

41 3


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414 H . FOSSEN and B. TIKOFF(e .g . F l inn 1979). Fo r p lane s t ra in th i s mat r ix t ran s fo rm -a t i o n is e q u a l t o t h e t r a n s fo rm a t i o n e q u a t i o n s

x ] = D l l x l + D 1 2 x 2X ~ = D 2 1 X 1 -4 - D 2 2 x 2 ,

w h e r e D ij a r e t h e c o m p o n e n t s o f D , a n d w h e r e a n yt r a n s l a t i o n i n v o l v e d i s n e g l e c t e d . Kn o wi n g t h e d e fo r -m a t i o n (g ra d i e n t ) m a t r i x D , t h e o r i e n t a t i o n a n d g e o m -e t ry o f t h e s t ra i n e l l i p s e a r e e a s i l y c a l c u la t e d ( s e e t h eAp p e n d i x ) . T h e d e fo rm a t i o n m a t r i c e s fo r s i m p l e s h e a ra n d p u re s h e a r w i t h o r w i t h o u t d i l a ti o n a r e :

Dss = ' Dps'~ = k2 'wh e re 7 i s th e s h e a r s t r a in , a n d k ~ a n d k 2 a r e t h ee x t e n s i o n -c o n t r a c t i o n a l o n g t h e X l a n d x 2 a x is , r e s p e c t -i v e ly . B y d e fi n i ti o n ( e . g . R a m s a y & H u b e r 1 9 8 3 ), k~ =1 / k 2 fo r p u re s h e a r , h e n c e i f k l k 2 # 1 , a v o l u m e c h a n g e isi n v o l v e d i n a d d i t i o n t o t h e p u re s h e a r i n t h e m a t r i xDp s , a . A f a m i l i a r , a n i s o t ro p i c v o l u m e c h a n g e c o m p a t -i b l e w i t h b o t h s i m p l e s h e a r a n d p u re s h e a r i s ( e . g .S a n d e r s o n 1 97 6, R a m s a y 1 9 8 0) :

1A = 1 + A ' (3)wh e re A is t h e v o l u m e c h a n g e . B e c a u s e t h e m a t r i c e s Opsa n d D a a r e b o t h d i a g o n a l , t h e p u r e s h e a r a n d v o l u m ec h a n g e c a n e a s i ly b e c o m b i n e d i n t o o n e m a t r i x b y s i m p l emat r ix mul t ip l ica t ion :

I ) ps , A = D p s D A = D a D p s

= [0 (1 + 0 A )k _ l ]= [k l k0 2] , ( 4 )w her e 1 + A = d e t Dps,a = k t k 2 . I n g e n e r a l, h o w e v e r ,m a t r i x m u l t i p l i c a t i o n i s n o n -c o m m u t a t i v e , a n d t h e s i m -u l t a n e o u s c o m b i n a t i o n o f s i m p l e s h e a r i n g a n d p u reshear ing -d i la t ing in to a s ing le , un i f ied mat r ix i s no t as t r a ig h t f o rw a r d p r o b l e m :[ 1 ~ k O z ] [~ ] # [ ~ 7 ] r k l (5 )T h e l e f t -h a n d s i d e o f (5 ) i s , m a t h e m a t i c a l l y , a s i m p l es h e a r d e f o r m a t i o n f o l l o w e d b y p u r e s h e a r a n d / o r v o l-u m e c h a n g e , a n d t h e r i g h t -h a n d s i d e i s a p u re s h e a r a n d /o r v o l u m e c h a n g e f o l l o w e d b y si m p l e s he a r . B o t h t h ed e fo rm a t i o n p a t h a n d t h e f i n i t e s t r a i n a r e d i f f e r e n t fo rt h e t wo d i f f e r e n t o rd e r s o f d e fo rm a t i o n s , a n d d i f f e r e n tf ro m s i m u l t a n e o u s l y a c t i n g p u re s h e a r i n g a n d s i m p l eshear ing .

In g e o l o g y , t h e f i n i t e s t r a i n i s c o m m o n l y k n o wn , a n dt h e re fo re f i x e d . An y f i n i t e d e fo rm a t i o n , wh e re o n l y t h ein i t ia l and f ina l pos i t ions o f po in t s o r vec to rs (x ) a rec o n s i d e re d , c a n b e f a c t o r i z e d i n a n i n f i n i t e n u m b e r o fw a y s in t o t w o o r m o r e d e f o r m a t i o n s , e a c h r e p r e s e n t e db y a s i ng l e d e fo rm a t i o n m a t r i x . A c o m m o n f a c t o r i z a t io ni s d e c o m p o s i t i o n i n t o ro t a t i o n a n d s t r e t c h (E l l i o tt 1 9 7 2 ) ,a n o t h e r i s f a c t o r i z a t i o n i n t o s im p l e s h e a r , p u re s h e a r

X 21 ( ................~0 '

~ p , ' ~ p , = ta n ~ = 4 . 3 3

a ) P ure shear ing , t hen s imp le sh ear ing

b

7,.p =1 .08

'1) S imp le shea r ing , t hen pure shea r ing

1

CC

~ F = l ~ l t a n ~ = 2 . 1 6

| i |'1 7 =k =2- - k ") S imu l t aneous s imp le an d pure shear ingF i g . 1 . T h r e e w a y s to a r r i v e a t t h e s a m e f i n i t e s t a t e o f d e f o r m a t i o n .( a ) A p u r e s h e a r i s f o l l o w e d b y a s i m p l e s h e a r . ( b ) A s i m p l e s h e a r i sf o l l o w e d b y a p u r e s h e a r , a n d ( c ) a s i m u l t a n e o u s p u r e s h e a r i n g a n ds i m p l e s h e a r i n g . N o t e t h a t t h e r e s p e c t i v e s h e a r s t ra i n s i n v o l v e d h a v eb e e n c h o s e n s o t h a t t h e s a m e f i n i t e s t r a i n i s a c h i e v e d i n a l l c a s e s ,w h e r e a s t h e p u r e s h e a r c o m p o n e n t i s t h e s a m e ( k = 2 ) i n al l c a s es .

and /o r d i la t ion , as d i scussed in th i s a r t i c le (F ig . 1 ) .A l t h o u g h a n y s u c h f a c t o r i z a t i o n i s , m a t h e m a t i c a l l y , as e q u e n c e o f l i n e a r t r a n s fo rm a t i o n s (d e fo rm a t i o n s ) , t h e yh a v e b e e n u s e d m e re l y a s a wa y o f r e p re s e n t i n g t h e f i n it es t r a i n i n a s h e a r z o n e b y t wo d i f f e r e n t p a ra m e t e r s .Ho we v e r , t h e s h e a r s t r a i n o n e a c h s i d e o f (5 ) m u s t b ed i f f e r e n t fo r t h e f i n i te s t r a in t o b e t h e s a m e :

wh ere 7p ,s # Ys.p fo r non -zero shear s t ra in va lues .F o r (6 ) t o h o l d t ru e , t h e r e l a t i o n s h i p b e t w e e n t h e t wo

s h e a r s t r a i n s m u s t b e7p,s = ( T s , p ) ( k x /k 2 ) . (7 )

I t may b e no ted tha t 7p ,s = tan (~p) (see F ig . 1 ) , and thefa c t o r i z a t io n o n t h e l e f t -h a n d s i d e o f (6) i s t h e r e fo rec o m m o n l y p re f e r r e d ( e . g . S a n d e r s o n 1 9 8 2) .

An y f a c t o r i z a ti o n o f t h e d e fo rm a t i o n m a t r i x c a n b eu s e d fo r t r e a t i n g f i n i t e s t r a i n ( s h a p e a n d o r i e n t a t i o n o fthe s t ra in e l l ip so id ) and f in i te ro ta t ion and change o fl e n g t h o f m a t e r i a l l i n es (p a s s iv e m a rk e r s ) . H o w e v e r , i fone i s in te res ted in the s t ra in h i s to ry , a s t ra in fac to r iza-t ion has s ign i f ican t impl ica t ions . For example , anin i t i a l ly a rb i t ra r i ly o r ien ted l ine wi l l ro ta te to the exac ts a m e p o s i t i o n a n d e x p e r i e n c e t h e s a m e f i n i t e e x t e n s i o ni r r e s p e c t i v e o f t h e s t r a i n f a c t o r i z a t i o n c h o s e n , b u t t h es t ra i n h i s t o ry o f t h e l in e ( i . e . s h o r t e n e d , t h e n e x t e n d e d ,e t c . ) w i l l b e d i f f e r e n t . I n f a c t , t o g e t h e r w i t h s o m e


3/10

T h e d e f o r m a t i o n ma t r i x a n d t r a n s p r e s s i o n - t r a n s t e n s i o n 4 15s impl i fy ing assumpt ions , th i s can be u t i l i zed to ex t rac tt h e v o r t i c i ty ( s e e b e l o w ) f r o m d e f o r m e d r o c k s , a s o u t -l ined by Passch ier (1990). S ince an a rb i t ra ry fac to r i za-t i o n o f d e f o r ma t i o n p r e v e n t s a r e a l is t ic mo d e l l i n g o f t h ed e f o r m a t i o n h i s to r y , a n d s i n ce p r o g r e ss i v e d e f o r ma t i o ng e n e r a l l y i n v o l v e s s i mu l t a n e o u s s i mp l e s h e a r i n g a n dp u r e s h e a r i n g , a d e f o r ma t i o n ma t r i x w h i c h c o mb i n e ss i mu l t a n e o u s p u r e s h e a r i n g a n d s i mp le s h e a r i n g is o f t e nn e e d e d . Su c h a d e f o r ma t i o n ma t r i x w a s d e r i v e d b yRa mb e r g ( 1 97 5, e q u a t i o n 3 8 ) i n a c o n t i n u u m me c h a n i c sf r a m e w o r k :( x ~ l = ( e x p lk x , t ) ? e x p ( ~ x l t ) - e x p ( - ~ x ' t ) ]2 k x , ( X l ] , ( 8 )\ x ~ J e x p ( - k x l t \X2Jw h e r e k x l i s the ra t e o f pure shear ing , i . e . t he ex tens ionra te para l l e l t o the sh ear d i rec t ion , ~ i s the ra t e o f s imples h e a r i n g a n d t i s t i me . U s i n g c o n t i n u u m me c h a n i c sno m enc la tu r e (M alvern 1969, Mea ns 1990) , kx, = L l la n d ~ = L 1 2, w h e r e L i j are th e ve loc i ty g rad ien t s .A l t h o u g h m a t r i x ( 8) m a y b e u s e d i n i ts p r e s e n t f o r m(e .g . R am be rg & G hosh 1977 , Ing les 1983 , Kl ig f ie ld e ta l . 1 98 5) , a s o m e w h a t s imp l e r , t i me - i n d e p e n d e n t v e r -s i o n o f ( 8 ) ma y b e d e r i v e d a s i n d i c a t e d b y Co w a r d &Kim (1981) and M er le (1986) :

k 7 ( k - k - ' )D = k - 1 = 0 k - t ' ( 9 )

w h e r e t h e o f f - d ia g o n a l ( r o t a t io n a l ) t e r m i s a f u n c t i o n o ft h e p u r e s h e a r i n g a n d s imp l e s h e a r i n g c o m p o n e n t s , a n dma y b e t e r m e d F ( e f f e c t iv e s h e a r s t r a in ) . H e r e , 7 = L 1 2 t ,a n d k = L i l t .T h e r e i s a u n i q u e o r i e n t a t i o n a n d g e o m e t r y o f t h ef in i t e s t ra in e l l ipse fo r a cer t a in s imul t aneous combi -n a t i o n o f s i mp l e s h e a r i n g a n d p u r e s h e a r i n g , a s s h o w n in

F i g . 2. T h e r e s u l t o f u s i ng ( 9 ) c a n b e a p p r o x i ma t e d b yp e r f o r m i n g a la r g e n u m b e r o f s u c c e ss i ve p u r e s h e a r a n ds i mp l e s h e a r i n c r e me n t s , a s s h o w n b y R a m b e r g ( 19 75 ).T h e n u m e r i c a l v a l u e o f 7 i n ( 9 ) is a l w a y s b e t w e e n t h o s eof 7p ,s and 7s ,p , and the i r re l a t ionsh ip i s:

7p,s = k F = k2(ys,p). (10)T h e e f f e c t o f v a r i o u s s i mu l t a n e o u s c o mb i n a t i o n s o fs i mp le s h e a r i n g a n d p u r e s h e a r i n g o n d i s p l a c e me n t a n dr o t a t i o n i n t h e s h e a r d i r e c t i o n c a n b e e x t r a c t e d f r o m ( 9 ) ,as i l lu s t ra t ed in F ig . 3 , w h ich a l so shows the re l a t ionsh ipbe tw een the s hea r s t ra in (Yp,s ) used b y Sa nderso n (1982)and th e shea r s t ra in (Y) fo r s imu l t aneous s imple shear inga n d p u r e s h e a r i n g . H e n c e , i f t h e g e o me t r y a n d o r i e n -t a t ion o f the s t ra in e l l ipse (R and 0 ' ) can be es t imatedf rom a shear zo ne , th en 7 , 7p , s, 7 s ,p , F an d k can b e fou ndby us ing Fig . 2 and /o r equa t ion (10) . I f t he angu lar shears t ra in in the Xa d i rec t ion OP) can be measured , thene i t h e r 7 o r 7 p,s c a n b e f o u n d f r o m t h e r e l a ti o n s h i p s

tan v2 = kF = Yp,~o r

(11)2 I n k7 = ~ Yp,s (12)

H e n c e , s i n c e k i s i n d e p e n d e n t o f t h esee Fig. 1) .d e f o r m a t i o n h i s t o r y , f i n d in g y f r o m 7 p , s or v ice versa i strivial.

T o a c c o u n t f o r d i l a t i o n , t h e u n i f i e d d e f o r ma t i o nma t r i x , D , b e c o m e s I k ~ Y ( k l - k 2 )( k F ] = l n ( k l / k 2 ) (13)D = k 2 k2

Fo r s i mu l t a n e o u s s i mp l e s h e a r i n g a n d d i l a t i n g p e r -p e n d i c u l a r t o t h e s h e a r z o n e , t h e r e l a t i o n s h ip b e t w e e nt h e o r i e n t a t i o n o f t h e s t r a in e l l ip s e a n d t h e s h e a r p l a n e(shear zo ne bo und ary ) is g iven by Fig . 4 . Th i s s i tua t ion

I~ o I , - o e k = & I ~ '= 0.25

70 - 1.5

O ! . , = ., o

" " . . . . . . 1 . 5 ~ ~

0 ~ 1 J , 5 . . . . 1 " 0 2 " 0 3 0 , 4 0 g O " " " 1 0 0R = 1 / ~ ' 1 / ~ , 2

F i g . 2 . R -O ' d i a g r a m w h e r e 8 ' i s t h e a n g l e b e t w e e n t h e s h e a r p l a n e ( z o n e ) a n d t h e l o n g a x i s o f t h e f i n i te s t r a i n e l l i p s e , a n d Ri s t h e s t r a i n a x i s r a t i o . Y p , s c o n t o u r s ( f r o m S a n d e r s o n 1 9 8" )) a r e s h o w n a s t h i c k , g r e y l i n e s .


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416 H . FOSSEN and B. TIKOFF

. . . . . . . . . . . .~ " o " k = l ' ' ' ' "

~ i I n p l e /

40 [ [ / / / ~ ~ ~ 1 / 1 ~ 0 . 1 / 2 o _ 1 ~ _ ~ .**, .**- 1/5.. '* 1/100 ~20 ~ * * * 1 / 1 0 0 0 ~

1 '~ 2 3 4F i g . 3 . V a r i a t i o n i n a n g u l a r s h e a r s t r a i n ( ~ ) f o r s i m u l t a n e o u s s i m p l e s h e a r in g a n d p u r e s h e a r in g . T h e d i s p l a c e m e n t ac ro s ss u c h a p u r e s h e a r - s i m p l e s h e a r z o n e = f i n i te v e rt i c a l d i s t a n c e - t a n ~ p.

wa s t r e a t e d i n d e t a i l b y R a m s a y (1 9 8 0 , e q u a t i o n s 1 7 -2 4 )fo r f in i t e s t ra in , who based h i s d i scuss ion on fac to r iza-t i o n o f t h e f i n it e d e fo rm a t i o n a s s i m p l e s h e a r fo l l o w e d b yv o l u m e c h a n g e (1 4 ) :I 0 0 f O o0 0 1 01 0 10 I + A 0 1 1 1 0 1 0 0 0 I + A (14)I f o n e i s i n t e r e s t e d i n t h e d e fo rm a t i o n h i s t o ry , h o w e v e r ,t h e d e fo rm a t i o n m a t r i x fo r s i m u l t a n e o u s s i m p l e s h e a r -i n g a n d p ro g re s s i v e v o l u m e c h a n g e , wh i c h c a n b e d e -r i v e d f ro m (9 ) t o b e

1 0 In (1 + A) (1 5 )0 1 0 '0 0 I + A

i s m o re u s e fu l . As a b o v e , t h e re i s a d i f f e r e n c e b e t we e n7s ,a in (14 ) and the 7 in (15 ) . Subs t i tu t ing the re la t ion -sh ip

A7s,A -- In (1 + A) 7 (16 )i n t o R a m s a y ' s f o rm u l a s g i v e s t h e c o r r e s p o n d i n g fo rm u -l a s fo r s i m u l t a n e o u s s i m p l e s h e a r i n g a n d p ro g re s s i v ev o l u m e c h a n ge .

T H E T H R E E - D I M E N S I O N A L D E F O R M A T I O NM A T R I XT h e d i s c u s s i o n a b o v e c a n b e e x t e n d e d t o t h r e e d i m e n -

s i o n s, a n d we w i l l d i s c us s t h e s i m u l t a n e o u s c o m b i n a t i o n

,

904 5

70

6050

4 0

3 02 0

10

0'1 n , ~ 1 / , ~ ~ ' = ' ~ ' % 2 3 4 s 6 7 8 9 1 0

F i g . 4. S i m i l a r t o F i g . 2 b u t f o r s i m u l t a n e o u s s i m p l e s h e a r a n d v o l u m e c h a n g e . N o t e t h a t t h i s d i a g r a m i s d i f f e r e n t f r o m as i m i l a r d i a g r a m s h o w n b y K l i g f i e l d e t a l . ( 1 9 8 1 , f ig . 12 ) a n d R a m s a y & H u b e r ( 1 9 8 3 , p . 5 0 ) m e r e l y b e c a u s e o f t h e d i f f e r e n tm e a n i n g o f y in t h e s e w o r k s .


5/10

T h e d e f o r m a t i o n m a t r i x a n d t r a n s p r e s s i o n - t r a n s t e n s i o n 4 1 7xa

x~ x2

2F i g . 5 . T h e d e f o r m a t i o n s a c c o u n t e d f o r i n t h e t h r e e - d i m e n s i o n a ld e f o r m a t i o n m a t r i x ( e q u a t i o n 1 7 ) .

o f t h r e e o r t h o g o n a l s i m p l e s h e a r s y s t e m s a n d a t h r e e -d i m e n s i o n a l p u r e s h e a r w i t h o r w i t h o u t a d d i t i o n a l v o l-u m e c h a n g e . T h i s g e n e r a l d e f o r m a t i o n c o v e rs a n u m b e ro f r e al i s ti c d e f o r m a t i o n s in t h e c r u s t , a n d c a n b e e x -p r e s s e d b y t h e m a t r i x

[k~ F 1 2 F13 ]D = k 2 1-'23 [ ,

0 k3 Jw h e r e t h e

(17)

k ~ , k2 a n d k 3 r e p r e s e n t t h e e x t e n s i o n s -c o n t r a c t i o n s a l o n g t h e X l , x 2 a n d x 3 c o - o r d i n a t e a x e s( i n c l u d e s v o l u m e c h a n g e i f k l " k 2 " k 3 ~ 1 ) , a n d t h eo f f - d i a g o n a l t e r m s ( Fq ) r e p r e s e n t e l e m e n t s o f s h e a rd e f o r m a t i o n . I f w e c o n s i d e r a le f t - h a n d e d c o - o r d i n a t es y s t e m , F 12 r e f l e c t s a w r e n c h i n t h e X l d i r e c t i o n , a n d F 13a n d F 23 c o r r e s p o n d t o a t h r u s t i n t h e X l a n d x 2 d i r e c t i o n s ,r e s p e c t i v e l y ( F i g . 5 ) .

T h e d e f o r m a t i o n c o m p o n e n t s i n v o l v e d c a n b e i l l u s -t r a t e d b y t h e i r i n d i v id u a l d e f o r m a t i o n m a t r i c e s :0 0 0 ] i1 0 1k2 , 0 1 0 ,

0 0 k3 0 0 1p u r e s h e a r _+ w r e n c h i n t h ev o l u m e c h a n g e x~ d i r e c t io n( 1 o [ 1 o ) 10 1 0 , 0 1 7

0 0 1 0 0t h r u s t i n t h e t h r u s t i n t h eX l d i r e c t i o n x 2 d i r e c t i o n

(18)

I t c a n b e s h o w n ( T i k o f f & F o s s e n i n p re s s) t h a t t h es i m u l t a n e o u s c o m b i n a t i o n o f t h e s e d e f o r m a t i o n s g i v e st h e d e f o r m a t i o n m a t r i x ( 1 7 ) w i t h

F12 - yw (k l - k2) (19a)In ( k l / k 2 )15z3/5-L

F13 _ ) 'Tl (kl -- k3) + 7 T 2 7 w ( k 1 - - k2 )In ( k l / k 3 ) In ( k l / k 2 ) In ( k 2 / k 3 )

+ 7T2) 'w(k3 - kl ) (19b)In ( k 2 / k 3 ) In ( k i / k 3 )F 2 3 - ) 'T2(k2 -- k3) (19c )In ( k 2 / k 3 )

W i t h ( 1 9 a ) - ( 1 9 c ) , m a t r i x ( 1 7 ) c o n t a i n s a l l t h e i n f o r -m a t i o n a b o u t t h e d e f o r m a t i o n , s u c h a s t h e o r i e n t a t i o n s ,m a g n i t u d e s a n d r o t a t i o n s o f t h e p r i n c ip a l d i r e c t i o n s ins p a c e .

T h e c a l c u l at i o n o f s tr a i n p a t h s i n t h r e e d i m e n s i o n s i ss i m i l a r t o t h e i r c a l c u l a t i o n i n p l a n e s t r a i n , a s s h o w na b o v e . T h e d e f o r m a t i o n m a t r i x ( 1 7) a b o v e i s n o t r e -s t r i c t e d t o c o m b i n a t i o n s o f t h r u s t i n g i n t h e x l a n d x 2d i r e c ti o n s a n d w r e n c h i n g i n t h e x l d i r e c t i o n , i n a d d i t i o nt o t h e g e n e r a l c o a x i a l s t r a i n a n d / o r v o l u m e c h a n g e .S w i t c h i n g t h e o r i e n t a t i o n s o f X l a n d x 2 g i v e s a c o m b i -n a t i o n o f t h r u s t i n g i n t h e x l a n d x 2 d i r e c t i o n s a n dw r e n c h i n g i n t h e x 2 d i r e c t i o n . S w i t c h i n g x l a n d x 3( m a k i n g x 1 v e r t i c a l ) g i v e s a c o m b i n a t i o n o f t w o v e r t i c a ls h e a r s a n d w r e n c h i n g i n t h e x 2 d i r e c t io n . M a k i n g x 2 t h ev e r t i c a l d i r e c t i o n g i v e s w r e n c h i n g a n d t h r u s t i n g i n t h e x ld i r e c t i o n a n d v e r t i c a l s h e a r i n t h e x 2 - x 3 p l a n e . H e n c e ,t h e e x a c t d e f o r m a t i o n m a t r i x f o r a l a r g e v a r i e t y o fc o m p l e x d e f o r m a t i o n s c a n b e f o u n d u s i n g ( 1 7 ) a n d( 1 9 a ) - ( 1 9 c ) . T h e r e s t r i c t io n s a r e th a t t h e d e f o r m a t i o n i sh o m o g e n e o u s , t h a t t h e c o a x i a l p r i n c i p a l d e f o r m a t i o na x e s ar e p e r p e n d i c u l a r t o t h e s h e a r p l a n e s , a n d t h a t t h es h e a r p l a n e s a r e m u t u a l l y o r t h o g o n a l .

V O R T I C I T YT h e k i n e m a t i c v o r t i c it y n u m b e r , W k ( T r u e s d e l l 1 9 5 3 ),

i s a u s e f u l m e a s u r e o f n o n - c o a x i a l i t y , p a r t i c u l a r l y i nc a s es w h e r e i t c a n b e e x t r a c t e d f r o m n a t u r a l l y d e f o r m e drocks (e .g . Pas sch ie r 1987 , 1990 , Pas sch ie r & Ura i 1988 ,V i s s e r s 1 9 8 9 ), a n d i s d i s c u s s e d b y M e a n s e t a l . (1980) . I t sr e l a ti o n s h i p w i t h s i m u l t a n e o u s c o a x i al d e f o r m a t i o n a n ds i m p l e s h e a r i n g i s

Wk = 7{2(In kl) 2 + 2(ln k2) 2 + ) ,2}-1/2 (20)f o r t w o d i m e n s i o n s ( 1 0 ) , a n d

W k - - -- { ( r T 1 ) 2 -+ - ( r W ) 2 - {- ( ~ / T 2 ) 2 } 1/ 2 {2(In k l) 2 + 2(In k2) 2 + 2 ( I n k3) 2

+ ( Y T 1 ) 2 + ( Y T 2 ) 2 + ( y W ) 2 } - 1 / 2 (21)f o r t h r e e d i m e n s i o n s ( 17 ) ( f o r d e r i v a t i o n , s e e T i k o f f &F o s s e n i n p re s s ). C o m b i n a t i o n s o f c o a x i al d e f o r m a t i o na n d s i m p l e s h e a r in g ( s ) p r o d u c e v o r t ic i t y n u m b e r s b e -t w e e n 0 a n d 1 , w h e r e W k = 0 f o r p u r e s h e a r i n g a n d W k =1 f o r s i m p l e s h e a r i n g .

S T R A I N P A T H ST h e r e a r e a n i n f i n it e n u m b e r o f st r a in p a t h s t h a t c a n

p r o d u c e a p a r t i c u l a r s t a t e o f f i n it e st r a i n . T h e s i m p l e s t is


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418 H. FOSSEN and B. TIKOFFs t e ady fl ow , which has been assumed in this paper. Bysteady flow, we mean that the incremental strain matrixdoes not change during the deformation history, i.e. theprincipal instantaneous stretching directions and thekinematic vorticity number (Wk) stay constant. Thestrain pa th can then be studied by choosing a fixed strainincrement and calculating the deformation matrix aftereach increment. For n increments and steady flow defor-mation, the incremental deformation matrix for a simul-taneous pure shearing, simple shearing and dilating canbe written in terms of the total pure shear and simpleshear components as

r i . c r lD i n c r = k o c , l i . k2,,J 1/n 1/nl /n , , -1 . (k , ,o) ]~, ' 1 ,o , / }* to t " ln--~l, Tk~ [" (22)o o !

0 ( k a t o t ) l / n JMatrix (22) gives the exact incremental strain for anysteady flow formed by a simultaneous combination ofpure shearing and simple shearing. A similar matrix canbe found for three-dimensional strain by using the re-lationships 7incr (Ytotal)/n, and k i n c r = (ktotal) l/n). Therelationship between the incremental and finite defor-mation matrix for stready flow is

X 2Fi g . 6 . Sc h e ma t i c i l l u s t ra t i o n o f t r a n sp re s s i o n a s a c o mb i n a t i o n o fs i m u l t a n c o u s s i m p l e s h e a r a n d p u r e s h e a r . B a s e d o n d i a g r a m b ySa n d c rso n & M a rc h i n i (1 9 8 4 ) .

Sanderson & Marchini (1984), in their discussion onf inite deformations in transpression zones, factorized thetotal deformat ion matrix into a pure shear and a simpleshear component:

D= 0 1 0 0 k0 0 1 1 0 k =I

O t o ta = ( D i n c r ) n . ( 2 3 )T h e o r i e n t a t io n s , m a g n i t u d e s a n d r o ta t i o n s o f t h e p r i n -c ipa l s t ra ins can be calcu lated prec ise ly at any s tep (seet h e Ap p e n d i x ) , a n d t h u s m a p p e d t h r o u g h o u t t h e d e f o r-m a t i o n .No n - s t e a d y f l o w d e f o r m a t i o n p a t h s c a n b e m o d e l e db y c h a n g i n g t h e in c r e m e n t a l d e f o r m a t i o n m a t r i x g r a d u -a l l y . For examp le , i f the p rogress ive vo lume changed e m o n s t r a b l y h a d a d e c re a s in g r o le d u r i n g a s h e a ri n gevent, the incremental matrix can be changed graduallythroughout the matrix pre-multiplication process. How-ever, it is usually hard to obtain information about theflow history of a particular deforma tion, and steady flowmay be considered a reasonable 'standard of reference'for geological modelling (cf. Passchier 1990).

APPLICATION TO TRANSPRESSION-TRANSTENSION

= k 0 . (24)0 k -1Whereas such a factorization is a good choice for finite

deformations, the progress ive deformation involved intranspression-transtension zones is easier and moreaccurately modeled by using a simplified version ofmatrix (17):

1 F 0 [ 1 7 ( l - k )In (k -t ) 0D = [ [ 0 k 0 = [ 0 k 0 ' ( 2 5 )0 0 k -1 0 0 k -1

where k is the horizontal pure shear component in the x2direction (perpendicular to the shear plane) and y is theshear strain in the xl direction (horizontal). k > 1 and7 # 0 gives transtension , whereas k < 1 and y 0 givestranspression.

Transpression and transtension are two closely re-lated types of deformation which, by definition, involvethe simultaneous combination of simple shearing andpure shearing. If we adopt the constraints used bySanderson & Marchini (1984) (free upper horizontalsurface, fixed lower horizontal surface, no volumechange), transpression-transtension can mathemat-ically be modelled as a combinat ion of a vertical simpleshear (wrench) in the xl direction and a pure shear in thex 2 - x 3 plane (Fig. 6). The compatibility problem nor-mally involved with such deformation is reduced byletting the free upper surface represent the surface of theEarth.

Stra in geometryAny simultaneous combination of the simple shearingand pure shearing in the system indicated in Fig. 6 gives

rise to a unique state of strain. The shape of the strainellipsoid can be illustrated on a contoured, logarithmicFlinn diagram (Fig. 7), and it is clear tha t transtensionaldeformation gives rise to constrictional strain, whereastranspressional deformation results in flattening strain.This important fact, which was also pointed out bySanderson & Marchini (1984), predicts S(L)-dominatedfabrics in transpressional shear zones, and L(S)-fabricswhere the deformation is transtensional. Note, how-


7/10

T h e d e fo rm a t i o n m a t r i x a n d t r a n s p re s s i o n - t r a n s t e n s i o n 4 1 9

2 0 , , , , " , , " ~ , , ,

10

, , ; , , , , , , , , . , , , , , , , , , , , ,#

I I , / , , ' , , - . , " / . , . " ,

1 "~b = v ~ - z / ~ , a ~ 0 2 0 6 0

F i g . 7 . L o g a r i t h m i c F l i n n d ia g r a m c o n t o u r e d f o r y ( s i m p l e s h e a r ) a n dk ( p u r e s h e a r ) v a l u e s , u s i n g d e f o r m a t i o n m a t r ix ( e q u a t i o n 2 5 )( t r a n s p r e s s i o n - t r a n s t e n s i o n ) . P u r e s h e a r ( k ) c o n t o u r s a r e s i m i l a r t ot h o s e s h o w n b y S a n d e r s o n & M a r c h i n i ( 1 9 8 4 , fi g . 2 ) , b u t s i m p l e s h e a rc o n t o u r s ( y ) a r e d i f f e r e n t b e c a u s e o f d i f f e r e n t d e f in i t i o n s o f 7 ( s e ed i s c u s s i o n i n t e x t ) .

e v e r , t h a t a s i m i la r F l in n d i a g ra m p re s e n t e d b y S a n d e r -s o n & M a rc h i n i i s d i f f e r e n t f ro m F i g . 7 b e c a u s e t h e s h e a rs t r ai n u s e d b y S a n d e r s o n & M a rc h i n i ( h e re c a l l e d Yp,s) i sd i f f e r e n t f ro m o u r 7 .

T h e t h r e e - d im e n s i o n a l orientation of the s t ra in e l l ip sec a n b e m a p p e d f o r v a r io u s c o m b i n a t i o n s o f p u r e s h e a ra n d s i m p l e s h e a r c o m p o n e n t s (F i g . 8 ) . 22 i s v e r t i c a l f o rw r e n c h i n g d e f o r m a t i o n s d o m i n a t e d b y s i m p le s h e a r in g .H o w e v e r , f o r h ig h l y t r a n s p re s s i o n a l d e fo rm a t i o n , t h e ;t~ax is is ver t i ca l , w her eas ) .3 i s the v er t i ca l p r inc ipa l s t ra ina x i s fo r h i g h l y t r a n s t e n s i o n a l d e fo rm a t i o n . F o r s t e a d yf low, and Wk ~< 0 .81 , e i ther ; t l ( fo r t ransp ress ion ) o r 23(fo r t rans tens ion) i s the ver t i ca l p r inc ipa l ax i s th rough-o u t d e fo rm a t i o n (F i g. 8 ) . Ho we v e r , f o r 1 > W k > 0 .8 1 ,

x

X 2

90 0~ o " ~ _ ~ ~

~ ~ I

8 0 - ~ . ~ ~ 1 - 3 o. . ~ . . . . . . . . . . . . . . . . . . ~ _ _ _ _ T _ _ [

0 4 0 - - ~ W R E N C H I N G ~3o- ~ ~ . . ' " V ~6oE ~ ' . ' " -' ~ ~ . . ' ~ [2 0 - ~ ~ . , " , , 0 l 7 0.,.., . ." ~ I"

, , ' '

0 " ' . , , , , , - , , , . . . . . I 9 00 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0

p u r e W k s i m p l es h e a r s n e a rF i g . 9 . Orientationofthemaximuminstantaneousstretchingdirection( k l) w i t h v a r y i n g k i n e m a t i c v o r t i c it y n u m b e r , f o r t r a n s p r e s s i o n a n dt r a n s t e n s i o n . A t W k = 0 . 8 1 , t h e m a x i m u m i n s t a n t a n e o u s s t r e tc h i n gd i r e c t io n b e c o m e s v e r t i c a l f o r t r a n s p r e s s i o n , a n d t h e o r i e n t a t i o n o f ki s s h o w n a s a d a s h e d l i n e w h e r e e l i s v e r t i c a l .

22 s ta r t s ou t as the ver t i ca l p r inc ipa l s t ra in ax i s , bu tswi tches po s i t ion w i th 2x ( fo r t ra nsp res s ion ) o r ~-3 ( fo rt r a n s t e n s i o n ) d u r i n g d e fo rm a t i o n a c c o rd i n g t o t h e c o n -s tan t vo r t i c i ty pa ths in F ig . 8 .

T h e t wo p r i n c i p a l s t r a i n a x e s i n t h e h o r i z o n t a l p l a n ea re o b l i q u e t o t h e x a a n d x 2 a x e s , d e p e n d i n g o n t h ere l a t iv e a m o u n t s o f p u re s h e a r a n d s i m p l e s h e a r. T h eang le 0 ' tha t the longe s t o f the ho r izon ta l p r inc ipa l s t ra ina x e s m a k e s w i t h t h e X l ( s h e a r ) d i r e c t i o n i s gi v e n b y t h ee q u a t i o n

10

4-

0 -0.1 k 0 . 5 1 2 5Transpression ~ Transtension 10F i g . 8 . O r i e n t a t i o n o f t h e f i n i te s tr a i n e l l i p s e f o r t r a n s p r e s s i o n -t r a n s t e n s i o n , a n d d e f o r m a t i o n p a t h s f o r c o n s t a n t v o r t i c i t y d e f o r -m a t i o n s i n k - -y s p a c e . N o t e t h e c h a n g e i n t h e v e r t i c a l p r i n c i p a l s t r a i na x i s f o r p r o g r e s s i v e d e f o r m a t i o n s w i t h Wk > 0 . 8 1 . W = k i n e m a t i cv o r t ic i ty n u m b e r , a n d Wk = 1 i n d i c a t e s t h e s i m p l e s h e a r i n g p a t h .

0 ' = ar cta n (2hrma~-k~ r2 - 1- . (26 )A p a r t i c u la r l y u s e fu l p i e c e o f i n fo rm a t i o n i s p ro v i d e d

b y t h e d i r e c t io n o f t h e i n s t a n t a n e o u s s t r e tc h i n g a x e s ,a n d i n p a r t i c u l a r t h e a n g l e ( c t ) b e t we e n t h e m a x i m u mi n s t a n t a n e o u s s t r e t c h i n g d i r e c t i o n ( e l ) a n d t h e s h e a rz o n e (F i g. 9 ) . I f t e n s io n g a s h e s a r e p r e s e n t i n t h et r a n s p re s s i o n - t r a n s t e n s i o n z o n e , t h e y m a y b e u s e d toe s t i m a t e t h e v o r t i c i t y n u m b e r W k o f th e d e fo rm a t i o n( t h e r e l a t iv e a m o u n t o f p u re s h e a r i n g a n d s i m p l e s h e a r -ing ) , u s ing the g raph shown in F ig . 9 . The same g raph i si n s o m e c a s e s a l s o a p p l i c a b l e t o l a rg e r s c a l e f e a t u re s ,s u c h a s e x t e n s i o n a l f a u l t s , d i k e s wa rm s y s t e m s , t h ru s tf a u l t s a n d fo l d a x e s fo rm e d i n t r a n s p re s s i o n -t r a n s t e n s i o n z o n e s . Ho we v e r , o n c e fo rm e d , a l l s u c hfe a t u re s w i l l r o t a t e d u r i n g fu r t h e r d e fo rm a t i o n , a s d i s -c u s s e d b e l o w .


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420 H. FOSSEN an d B. TIKOFFC h a n g e s i n a n g l e s a n d l e n g t h s

T h e f o r m u l a s p re s e n t e d b y S a n d e r s o n & M a r c h i n i( 19 8 4) m u s t b e m o d i f i e d w h e n u s i n g m a t r i x ( 2 5) f o re i t h e r f i n i t e o r p r o g r e s s i v e s t r a i n c o n s i d e r a t i o n s b y s u b -s t i t u t i n g f o r t h e i r s h e a r s t r a i n ( h e r e c a l l e d Y p,s) w i t h t h ee x p r e s s i o n :

_ 7 ( 1 - k ) _ V ( 2 7 )7p.s k in k -1 kF o r e x a m p l e , t h e o r i e n t a t i o n o f t h e v e c t o r o r t h e p a s si v el in e a f t e r d e f o r m a t i o n t h e n i s d e f i n e d b y t h e e q u a t i o n

c o t 9 ' = k - ] c o t @ + F k - 1- - k - t c o t ~ + y ( 1 - k ) (28)k I n ( k - l ) '

wh ere q~ i s t he in i t i a l ang le , and qY i s the new angleb e t w e e n t h e l i n e a n d t h e x t a x i s . S i m i l a r ly , t h e e x t e n s i o n( q u a d r a t i c e l o n g a t i o n ) o f a h o r i z o n t a l l in e is g i v e n b y t h ee q u a t i o n

2ho = (COS q~ + I" sin 9) 2 + k 2 sin e q~. (29 )A n y l i n e i n s p a c e ( n o t n e c e s s a r i l y h o r i z o n t a l ) w i t h

i n i t i a l d i r e c t i o n g i v e n b y t h e u n i t v e c t o r l = ( a , b , c )t r a n s f o r m s i n t o t h e n e w v e c t o r l ' w h e r e[ a ] ( a + b F ]' = b ' k b

c ' c / k(30)

a n d t h e a c t u a l q u a d r a t i c e l o n g a t i o n o f t h e l i n e isC22 = - ~ + b2 k 2 + ( a + bF) 2 . (31)

T h e a n g l e a b e t w e e n t h e l i n e a n d t h e X l a x i s ( s h e a rd i r e c t io n ) i s g i v e n b y t h e e q u a t i o na ta = a r c c o s . ~ . ( 3 2 )V A

T h e r e s u l t s ( F i g s . 1 0 a n d 1 1 ) i n d i c a t e t h a t h o r i z o n t a lm a t e r i a l l i n e s w i ll r e m a i n h o r i z o n t a l d u r i n g b o t h t r an s -p r e s s io n a l a n d t r a n s t e n s i o n a l d e f o r m a t i o n . A h o r i z o n t a ll i n e w i t h i n i t ia l o r i e n t a t i o n 9 0 t o t h e s h e a r i n g ( x t )d i r e c t io n r o t a t e s v e r y s i m i la r ly u n d e r s i m p l e s h e a r i n ga n d t r a n s p r e s s i o n a l d e f o r m a t i o n . O t h e r h o r i z o n t a ll in e s , h o w e v e r , r o t a t e m o r e s l ow l y t o w a r d s t h e x t a x isw i t h t r a n s p r e s s i o n t h a n w i t h s i m p l e s h e a r i n g . T r a n s -t e n s i o n m a k e s a l l h o r i z o n t a l l i n e s r o t a t e s l o w e r t h a nd u r i n g s i m p l e s h e a r i n g , a n d t h e l i n e s r o t a t e t o w a r d s ah o r i z o n t a l , a s y m p t o t i c l i n e w h i c h m a k e s a b o u t 2 4 w i t ht h e x a ax i s w h e n Wk = 0 . 7 5 . T h i s a n g l e i s d e f i n e d b y t h ea p o p h y s i s ( c f . B o b y a r c h i c k 1 9 8 6) o f th e f l o w w h i c h i s n o tp a r a l l e l t o t h e x 1 o r x 3 a x e s . T h i s a p o p h y s i s , w h i c h i sh o r i z o n t a l , i s t h e e i g e n v e c t o r c o r r e s p o n d i n g t o t h e l ar -g e s t e i g e n v a l u e o f t h e v e l o c i t y g r a d i e n t f i e l d ( e . g .B o b y a r c h i c k 1 9 8 6, T i k o f f & F o s s e n i n p r e s s ) . T h e a n g l eb e t w e e n t h is o b l i q u e f l o w a p o p h y s i s a n d Xl c a n b e s h o w nt o b e

a r c t a n ( ' n " t ( 3 3 )\ 7 /

135

120

I00

60

40

20a ) o

- - - W k = l (s imple shear ing)W k = 0 . 7 5 , transtensionWk=O.75, t r a n s p r e s s i o n

t i l lt t%%%% %t

1.0 2.0 3.0 4.0 5.0 6.0

T h e s e e q u a t i o n s c a n b e u s e d t o s t u d y th e r e o r i e n t a t i o no f f o l d a x e s a n d o t h e r l i n e a r s t r u c t u r e s t h a t b e h a v e a sp a s s iv e m a r k e r s , o r n e a r l y s o , d u r i n g tr a n s p r e s s i o n -t r a n s t e n s i o n . F o l d a x e s t h a t i n i t i a t e a t a n a n g l e t o t h es h e a r p l a n e ( x r x 3 p l a n e ) w i l l r o t a t e d u r i n g d e f o r -m a t i o n . F o r s i m p l e s h e a r i n g a l o n e , f o l d a x e s a r e w e l lk n o w n t o r o t a t e t o w a r d s x l w i t h in c r e a s in g s h e a r s t r a in .U s i n g ( 2 5 ) , w e c a n n o w s t u d y t h e b e h a v i o r o f f o ld s int r a n s t e n s i o n a l - t r a n s p r e s s i o n a l r e g i m e s .

F o u r i n i t i a l o r i e n t a t i o n s h a v e b e e n c h o s e n a sexamples : 090/00 , 135/00 , 090/45 and 135/45 . The k ine -m a t i c v o r t i c i t y n u m b e r , W k, h a s b e e n k e p t f i x e d a t 0 . 7 5t h r o u g h o u t t h e d e f o r m a t i o n . A s a m e a s u r e o f th e t o t a lt h r e e - d i m e n s i o n a l s t r a i n , w e h a v e u s e d t h e u n i t

x/~_es = -~ - Yo,w h e r e Y-o i s t h e n a t u r a l o c t a h e d r a l u n i t s h e a r ( c f. H o s -sack 1968).

135

120

lOO

6O

4O

2O

b ) o

J i l l

1 . 0

" - . ' - .2.0 3.0 4.0 5.0 6.0, ,

F i g . 1 0 . C h a n g e i n o r i e n t a t i o n o f p a s s i v e l y d e f o r m i n g l i n e s ( e . g . f o l da x e s ) f o r s i m p l e s h e a r w r e n c h i n g ( d a s h e d l i n e ) a n d t r a n s t e n s i o n -t r a n s p r e s s i o n . I n ( a ) t h e i n i t i al l i n e s a r e 1 3 5/ 00 a n d 0 9 0 / 0 0 , a n d i n ( b )1 3 5/ 45 a n d 0 9 0 / 4 5 . a = a n g l e b e t w e e n t h e l i n e a n d t h e s h e a r d i r e c t i o n( x l ) . S e e t e x t f o r d i s c u s s i o n .


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T h e d e f o r m a t i o n m a t r i x a n d t r a n s p r e s s i o n - t r a n s t e n s io n 4 21

Xl t~ 7 ,! :; , ' , , , ' i

o , @ i i J i ) i i , l ' l i l \ \" 6 . " / , i s # I # 1 . 1 . t i t

TRANSTENSION

X1~ Wk=l.0-

, ~ ' \ ~ N

, ; , ) ) !X 1

\ e 3

TRANSPRESSlON

= 0 . 7 5

e 2

Fig. 11. Stereographic illustration of the progressive rotation of passive line markers for transpression-transtension(Wk = 0.75) and sim ple shearing (wrenching). Arrow s indicate the resu lt of simple shear strains of 0.25 combinedsimultaneously with k = 1.1165456 (transpression), k- I = 1.1165456 (transtension), and k = 1 (simple shear) to perform therotations. The lengths of the arrows therefore indicate the rates of rotation in the different fields of the stereograms. Thetotal deformation paths o f the four lines discussed in Figs. 10(a) & (b) are shown, kl is the largest instantaneous stretchingdirection, and L e is the flow apophysis n the direction defined by the eigenvector corresponding to the largest eigenvalue ofthe velocity gradient tensor L. L e is parallel to x I and x 2 or sim ple shearing and transpression, respectively.

T h i s a n g l e , w h i c h v a r i e s f r o m 9 0 f o r p u r e s h e a r i n g t o 0 f o r s i m p l e s h e a r i n g , i s a l s o t h e v a l u e w h i c h t h e m a x i m u mh o r i z o n t a l p r i n c i p a l s t r a i n a x i s a p p r o a c h e s ( b u t n e v e rr e a c h e s ) w i t h i n c r e a s i n g s t r ai n . T h i s i m p l i e s t h a t f o rt r a n s t e n s i o n , f o r w h i c h s t r o n g l i n e a r f ab r i c s a r ee x p e c t e d , t h e r e i s a l w a y s a c o n s i d e r a b l e a n g l e b e t w e e nt h e s t r e t c h i n g l i n e a t i o n a n d t h e s h e a r d i r e c t i o n ( > 2 4 f o rW k = 0 .75 ) .

A l l l i n e s i n c l i n e d a w a y f r o m t h e x l - x 2 s h e a r p l a n er o t a t e m o r e s l ow l y d u r in g t r a n s t e n s i o n - t r a n s p r e s s i o nt h a n d u r i n g s i m p l e s h e a r i n g ( F i g. 1 0 b ) . F o r t r a n s t e n -s i o n , th e a n g l e a p p r o a c h e s t h e a s y m p t o t i c v a l u e g i v e n b y( 3 3 ) , w h e r e a s f o r t r a n s p r e s s i o n t h e i n c l i n ed l in e s s ta r t t or o t a t e a w a y f r o m X l t o w a r d t h e v e r t i c a l x3 a x i s a t s o m ep o i n t . T h i s c h a n g e i n r o t a t i o n d i r e c t i o n o c c u r s o n l y i nt r a n s p r e s s i o n w h e n , t 1 i s v e r t i c a l ( F i g . 8 ) , a n d i s d u e t ot h e f a c t t h a t 2 1 a n d 2 2 a r e r e l a t i v e l y c l o s e i n m a g n i t u d e .I t c a n b e s e e n f r o m t h e f l o w p a t t e r n i n F ig . 1 1 ( c ) t h a t t h ise f f e c t i s l a r g e s t f o r i n i t i a l l y s h a l l o w l y p l u n g i n g l i n e s i nt h e f ir s t ( a n d t h i r d ) q u a d r a n t ( s ) o f t h e s t e r e o g r a m .T h e s e c o n s i d e r a ti o n s d e m o n s t r a t e t h a t f o ld h in g e s a n do t h e r l i n ea r f e a t u r e s t h a t b e h a v e i n a p as s iv e m a n n e r c a nn e v e r b e c o m e p a r a l l e l to t h e s h e a r d i r e c t i o n i n t r a n s -t e n s i o n a l s h e a r z o n e s , a n d o n l y i f t h e y o r i g i n a t e d a sp e r f e c t l y h o r i z o n t a l l i n es in t r a n s p r e s s i o n a l s h e a r z o n e s .

C O N C L U S I O N SP r o g r e s s i v e , a s w e l l a s f i n i t e d e f o r m a t i o n , c a n b e

m o d e l l e d u s i n g a s in g l e m a t r i x , g i v e n i n t e r m s o f t h ep u r e s h e a r i n g ( k ) a n d s im p l e s h e a r i n g ( 2 ) c o m p o n e n t s .T h e s t r a in p a t h c a n b e c a l c u l a t e d u s i n g t h e r e l a t i o n s h i p s

7incr = () 'total)/n and kincr = (k to tal ) ( l /n) , w h e r e 7 a n d ka r e , r e s p e c t i v e l y , s i m p l e s h e a r a n d p u r e s h e a r f a c t o r s .T h e s e s o l u t i o n s g i v e i d e n t i ca l f i n it e d e f o r m a t i o n s a n dd e f o r m a t i o n p a t h s t o t h o s e d e s cr i b e d b y R a m b e r g ' ss t r a i n r a t e - a n d t i m e - d e p e n d e n t e q u a t i o n s , b u t a r es i m p l e r t o u s e .

A p p l i c a t i o n o f th i s st r a i n t h e o r y s h o w s t h a t t r a n s -p r e s s i o n a l d e f o r m a t i o n p r o d u c e s f l a t t e n i n g o r p l a n a rf a b r i c s , w h e r e a s t r a n s t e n s i o n a l d e f o r m a t i o n r e s u l t s i ns t r o n g l y l i n e a r a n d c o n s t r i c t i o n a l f a b r i c s . A s s u m i n gs t e a d y f l o w , a n y f i n i te s t a t e o f s t r a i n i s t h e r e s u l t o f au n i q u e c o m b i n a t i o n o f s im u l t a n e o u s p u r e s h e a r i n g a n ds i m p l e s h e a r i n g . T h e o r i e n t a t i o n s o f th e p r i n c i p a l a x e s o ft h e f i n i t e s t r a i n e l l i p s e d e p e n d o n t h e v o r t i c i t y a s w e l l a so n w h e t h e r t h e d e f o r m a t i o n i s h i g h ly t r a n s p r e s s i v e ,t r a n s te n s i v e , o r s i m p l e s h e a r - d o m i n a t e d w r e n c h i n g .F u r t h e r m o r e , t h e c h a r ac t e ri s ti c p a t t e r n s o f r o t a ti o n o fm a t e r i a l l i n e s ( e . g . f o l d a x e s a n d l i n e a t io n s ) a r e s h o w nt o b e s i g n i f i c a n t l y d i f f e r e n t f o r t r a n s t e n s i o n a n d t r a n -s p r e s s i o n . I n t h e f o r m e r c a s e th e l i n e s r o t a t e t o w a r d s t h ef l o w a s y m p t o t e t h a t i s o b l i q u e t o t h e X l a x i s , a n d w i t ht r a n s p r e s s i o n , n o n - h o r i z o n t a l l i n e s e v e n t u a l l y r o t a t et o w a r d s t h e v e r t i c a l ( x 3 ) a x i s ( v e r t i c a l f l o w a s y m p t o t e )a n d a w a y f r o m t h e s h e a r d i r e c t i o n . A l s o , s t r a i n h is t o r ie sd i f f e r e n t f r o m s t e a d y f l o w c o n d i t i o n s c a n b e m o d e l e du s i n g g r a d u a l l y c h a n g i n g i n c r e m e n t a l d e f o r m a t i o nm a t r i c e s .Acknowledgemen t s - - Th i s work was supported by the NorwegianResearch Council for Science and the H umanities (NAV F grant No.440.89/061) for H . Fo ssen , and a Stanwood Johnson fellowship(University of M innesota) for B. Tikoff. Reviews by W. M eans and D.Sanderson helped improve and clarify the m anuscript. P. Hudlestonand C. Teyssier are thanked for helpful conversations.


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422 H . FOSSEN an d B. TIKOFF

R E F E R E N C E S A P P E N D I XB o b y a rc h i c k , A. R . 1 9 8 6 . Th e e i g e n v a l u e s o f s t e a d y f l o w i n M o h rsp a c e . Te c t o n o p h y s i c s 122, 35-51 .C o w a rd , M . P . & Ki m, J . H . 1 9 81 . S t ra i n wi t h i n t h ru s t sh e e t s . In :T h r u s t a n d N a p p e T e c t o ni c s (e d i t e d b y M C l a y , K. R . & Pr i c e , N. J . ) .S p e c . Pu b l s g e o l . S o c . Lo n d . 9 , 2 7 5 -2 9 2 .C o w a r d , M . P . & P o t t s , G . J . 1 9 8 3. C o m p l e x s t r a i n p a t t e r n s d e v e l o p e d

a t f r o n t a l a n d l a t e r a l ti p s t o s h e a r z o n e s a n d t h r u s t z o n e s . J . S truct .G e o l . 5 , 3 8 3 - 3 9 9 .E l l i o t t , D . 1 9 7 2 . De fo rm a t i o n p a t h s i n s t ru c t u ra l g e o l o g y . Bu l l . g e o l .S o c . A m . 8 3 , 2 6 2 1 -2 6 3 5 .F l i n n , D . 1 97 9. T h e d e f o r m a t i o n m a t r i x a n d t h e d e f o r m a t i o n e l li p s o id .J . S truct . Geol . 1 , 2 9 9 - 3 0 7 .H o s s a c k , J . 1 96 8. P e b b l e d e f o r m a t i o n a n d t h r u s t i n g i n th e B y g d i n a r e a( S o u t h e r n N o r w a y ) . Te c t o n o p h y s i c s 5 , 3 1 5 - 3 3 9 .In g l e s , J . 1 9 8 3 . Th e o re t i c a l s t r a i n p a t t e rn s i n d u c t i l e z o n e s s i mu l -t a n e o u s l y u n d e r g o i n g h e t e r o g e n e o u s s i m p l e s h e a r a n d b u l k s h o r t e n -ing . J . S truct . Geol . 5 , 3 6 9 - 3 8 1 .Kl i g f i e ld , R . , C re sp i , J . , Na ru k , S . & Da v i s , G . H. 1 9 85 . Di sp l a c e m e n ta n d s t r a i n p a t t e r n s o f e x t e n s i o n a l o r o g e n s. Te c t o n i c s 3 , 5 7 7 - 6 0 9 .Kl i g f i e ld , R . , C a rm i g n a n i , L . & O we n s , W. H . 1 9 8 1 . S t ra in a n a l y s is o fa N o r t h e r n A p e n n i n e s h e a r z o n e u s in g d e f o r m e d m a r b l e b r e cc i a s . J .Struct . Geol . 3 , 4 2 1 -4 3 6 .M a l v e rn , L . E . 1 9 6 9 . In t ro d u c t i o n t o t h e Me c h a n i c s o f a C o n t i n u o u sM e d i u m . P r e n t i c e - H a l l , E n g l e w o o d C l if f s, N e w J e r s e y .M e a n s , W . D . 1 99 0. K i n e m a t i c s , s tr e s s , d e f o r m a t i o n a n d m a t e r i a lb e h a v i o u r . J . S truct . Geol . 12, 953-971 .M e a n s , W. D . , Ho b b s , B . E . , L i s t e r , B . E . & Wi l l i a ms , P . F . 1 9 80 .Vo r t i c i t y a n d n o n -c o a x i a l i t y i n p ro g re s s i v e d e fo rma t i o n s . J . S truct .G e o l . 2 , 3 7 1 -3 7 8 .M e r l e , O. 1 9 86 . Pa t t e rn s o f s t r e t c h t r a j e c t o r i e s a n d s t ra i n ra t e s wi t h i ns p r e a d i n g - g l i d i n g n a p p e s . Te c t o n o p h y s i c s 124, 211-222 .Pa ssc h i e r , C . W . 1 9 87 . S t a b l e p o s i t i o n s o f r i g id o b j e c t s in n o n -c o a x i a lf i o w --a s t u d y i n v o r t i c it y a n a ly s i s . J . S truct . Geol . 9, 679-690 .Pa ssc h i e r , C . W. 1 9 90 . R e c o n s t r u c t i o n o f d e fo r ma t i o n a n d f lo wp a r a m e t e r s f r o m d e f o r m e d v e i n s e t s. Te c t o n o p h y s i c s 180, 185-199 .Pa ssc h i e r , C . W. & U ra i , J . L . 1 9 88 . Vo r t i c i t y a n d s t ra i n a n a l y s is u s i n gM o h r d i a g r a m s . J . S truct . Geol . 10, 755-763 .R a m b e r g , H . 1 97 5. P a r t i c l e p a th s , d i s p l a c e m e n t a n d p r o g r e s s i v e s tr a i na p p l i c a b l e t o ro c k s . Te c t o n o p h y s i c s 28, 1 -37 .R a m b e r g , H . & G h o s h , S . K . 1 97 7. R o t a t i o n a n d s t r a i n o f l i n e a r a n dp l a n a r s t r u c t u r e s i n t h r e e - d i m e n s i o n a l p r o g r e s s i v e d e f o r m a t i o n .Te c t o n o p h y s i c s 4 0 , 3 0 9 -3 3 7 .R a m s a y , J . G . & G r a h a m , R . H . 1 97 0. S t r a i n v a r i a t i o n i n s h e a r b e l ts .Can. J . Earth Sci . 7 ,7 8 6 - -8 1 3 .R a m sa y , J . G . 1 9 80 . Sh e a r z o n e g e o m e t ry : a r e v i e w. J . S truct . Geol . 2 ,8 3 -9 9 .R a m s a y , J . G . & H u b e r , M . I . 1 9 8 3 . T h e T e c h n i qu e s o f M o d e r nS t ru c t u ra l G e o l o g y , Vo l u m e i: S t ra i n An a l y s i s . A c a d e m i c P r e s s ,L o n d o n .S a n d e r s o n , D . J . 1 97 6. T h e s u p e r p o s i t i o n o f c o m p a c t i o n a n d p l a n es t ra i n . Te c t o n o p h y s i c s 3 0 , 3 5 -5 4 .S a n d e r s o n , D . J . 1 98 2. M o d e l s o f s t r a i n v a r ia t i o n s in n a p p e s a n d t h r u s tsh e e t s : a r e v i e w. Te c t o n o p h y s i c s 8 8 , 2 0 1 -2 3 3 .S a n d e r s o n , D . J . & M a r c h i n i , R . D . 1 9 8 4 . T r a n s p r e s s i o n . J . S truct .G e o l . 6 , 4 4 9 - 4 5 8 .T i k o f f , B . & F o s s e n , H . I n p r e ss . S i m u l t a n e o u s p u r e a n d s i m p l e s h e a r :t h e u n i f i e d d e f o r m a t i o n m a t r i x . Te c t o n o p h y s i c s .Tru e sd e l l , C . 1 9 53 . Two me a su re s o f v o r t i c it y . J . Ra t i o n a l Me c h . A n a l .2. 173-217.Vi s sc r s , R . L . M . 1 9 89 . Asy m me t r i c q u a r t z c -a x is f a b r i c s a n d fl o wv o r t i c i ty : a s t u d y u s i n g ro t a t e d g a rn e t s . J . S truct . Geol . 11, 231-244 .Wa l l i s , S . R . 1 9 92 . Vo r t i c i t y a n a ly s i s i n me t a c h e r t f ro m t h e Sa mb a -g a w a B e l t , S W J a p a n . J . S truct . Geol . 14, 271-280 .

T h e d e f o r m a t i o n m a t r i x f o r p la n e s t r a i n ( 1 0) a n d t h r e e d i m e n s i o n s( 1 7 a n d 1 9 a - 1 9 c ) h a s b e e n t e s t e d n u m e r i c a l l y a g a i n s t a c o m p u t e rp r o g r a m w h i c h g i v e s a n a p p r o x i m a t e s o l u t i o n b y s u c c e s s i v e l y p r e -m u l t i p ly i n g s m a ll i n c r e m e n t s o f p u r e s h e a r a n d s i m p l e s h e a r ( s ) . T h es o l u t io n c o n v e r g e d t o w a r d s t h e d e f o r m a t i o n m a t r i x a s t h e s i ze o f t h ei n c r e m e n t s w a s d e c r e a s e d . T h e m a t h e m a t i c a l d e r i v a t i o n s o f t h e s ema t r i c e s a re sh o wn i n T i k o f f & Fo sse n ( i n re v i s i o n ) .T h e v o l u m e c h a n g e ( A ) i n v o l v e d i n t h e d e f o r m a t i o n i s si m p l yd e t (D) - 1 1 0 0 %, a n d i s n e g a t i v e fo r v o l u me d e c re a se a n d p o s i t i v ef o r v o l u m e i n c r e a s e . T o f i n d t h e g e o m e t r y o f t h e s t r a i n e l l ip s o i d fr o mt h e t w o - o r t h r e e - d i m e n s i o n a l u n i f ie d d e f o r m a t i o n m a t r ix , f o r m t h em a t r i x D DT. Th e e i g e n v a l u e s o f t h i s ma t r i x (a l wa y s 2 fo r 2 2ma t r i c e s , a n d 3 fo r 3 3 ma t r i c e s fo r g e o l o g i c a l l y re a l i s t i c d e fo r -ma t i o n s ) a re t h e q u a d ra t i c p r i n c i p a l s t r a i n ma g n i t u d e s , e .g . 21 = (1 +e l ) a , a n d t h e i r c o r r e s p o n d i n g e i g e n v e c to r s g i v e t h e d i r e c t io n s o f t h ep r i n c i pa l a x e s i n th e d e f o r m e d s t a te . F o r t w o d i m e n s i o n s t h e e i g e n -v a l u e s ( l e n g t h s o f t h e s t ra i n e ll i p se a x e s ) a re g i v e n b y t h e fo rm u l a

r 22 = + ~ + k 2 + ~ / - 4 ~ + ( F 2 + k~l + k~2) (A 1)2

a n d t h e c o r r e s p o n d i n g e i g e n v e c t o r s c a n b e e x p r e s s e d a sF -k2r le = ~ 2 + ~ - 2 ~ . ( A 2 )

F o r t h r e e d i m e n s i o n s t h e e i g e n v a l u e s a n d e i g e n v e c t o r s a r e m o r ee a s i l y so l v e d fo r n u me r i c a l l y . Th e a n g l e 0 ' b e t we e n t h e l a rg e s t p r i n c i -p a l s t r a i n a x i s a n d t h e sh e a r (x x) d i re c t i o n ! s0 ' = a rc c o s (e l l ) (A3 )

w h e r e e I I is t h e f i r st c o m p o n e n t o f t h e n o r m a l i z e d e i g e n v e c t o r o f D D Tc o r re s p o n d i n g t o 21 ( t h e n o rm a l i z e d fo rm o f a v e c t o r v i s V/(v Tv) / 2 ) . Inb o t h t w o a n d t h r e e d i m e n s i o n s , o n e c a n s t u d y t h e c h a n g e i n o r i e n -t a t i o n o f a n y l i n e f ro m i t s i n i t i a l o r i e n t a t i o n , g i v e n b y t h e u n i t v e c t o r !( m a d e u p o f t h e d i r e c t i o n c o s i n es o f t h e l i n e ) t o t h e n e w d i r e c t io n l' b yt h e t r a n s f o r m a t i o nl ' = D1. (A4)T h e a n g l e o f r o t a ti o n ( ~ ) o f t h i s li n e ca n b e f o u n d f r o m t h e f o r m u l a

Frc o s ~ - ~ / / ,~ r~ (A5 )a n d t h e q u a d r a t i c e x t e n s i o n (2 ) o f t h e l i n e is s i mp l y

2 = l 'T1 . ( A 6 )Th e a n g l e f l b e t w e e n t h e l a rg e s t p r i n c i p a l s t r a i n a x i s ( e 1) a n d t h e l i n e i s

f l = a rc c o s (e T l ' ) . (A7 )w h e r e e 1 i s t h e n o r m a l i z e d e i g e n v e c t o r c o r r e s p o n d i n g t o t h e l a r g e s te i g e n v a l u e (2 1 ) o f DD T. Th e n e w v e c t o r l ' h a s , i n g e n e ra l , a l e n g t hd i f f e r e n t f r o m u n i t y , b u t m a y b e n o r m a l i z e d t o r e v e a l t h e n e wd i re c t i o n c o s i n e s w i t h re sp e c t t o t h e x 1 , x 2 a n d x 3 c o -o rd i n a t e a x e s ,re sp e c t i v e l y .S i mi l a r l y , if p i s t h e p o l e t o a p l a n e p r i o r t o d e fo r ma t i o n , t h e n e wo r i e n t a t i o n o f t h e p l a n e i s g i v e n b y p ' , w h e r e

p ' = p O - 1 . ( A 8 )T h e r o t a t i o n o f p e q u a ls t h e r o t a t i o n o f t h e p l a n e , a n d c a n b e f o u n d b yu s i n g e q u a t i o n ( A 5 ) .

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